Probing Magnetic Moment Operators in $H \gamma$ Production and $H \to \tau^+ \tau^- \gamma$ Rare Decay

The magnetic moment ($a_\gamma$) and weak magnetic moment ($a_W$) of charged leptons and quarks are sensitive to quantum effects of new physics heavy resonances. In effective field theory $a_\gamma$ and $a_W$ are induced by two independent operators, therefore, one has to measure both the $a_\gamma$ and $a_W$ to shed lights on new physics. The $a_W$'s of the SM fermions are measured at the LEP. In this work, we analyze the contributions from magnetic and weak magnetic moment operators in the processes of $pp\to H \gamma$ and $gg\to H \to \tau^+ \tau^- \gamma$ at the High-Luminosity Large Hadron Collider. We demonstrate that the two processes could cover most of the parameter space that cannot be probed at the LEP.